Theorem imaeq2 5125

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5060	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 5013	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4789	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4789	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ( C " A ) = ( C " B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   ran crn 4777   |` cres 4778   "cima 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-cnv 4784  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789

This theorem is used by:  imaeq2i  5127  imaeq2d  5129  fimadmfo  5627  ssimaex  5767  ssimaexg  5768  isoselem  6029  f1opw2  6299  supp0cosupp0fn  6511  fopwdom  7140  ssenen  7156  fiintim  7242  fidcenumlemrk  7276  fidcenumlemr  7277  sbthlem2  7280  isbth  7289  ennnfonelemp1  13355  ennnfonelemnn0  13371  ctinfomlemom  13376  ctinfom  13377  tgcn  15434  iscnp4  15444  cnpnei  15445  cnima  15446  cnconst2  15459  cnrest2  15462  cnptoprest  15465  txcnp  15497  txcnmpt  15499  metcnp3  15737